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4 Nov 2015: n 1)(n (n 1) cos x). (2n 1)(2n 1). xπ. 1. ... H′3 H′′′3 = G1. 1. Pn1 (1 cos x)n1, H3 Sx.

28 Jun 2016: x= a sin σcosh τ cos σ , y=. a sinh τcosh τ cos σ , (2.9a,b). ... n=1. enτ cos nσ. ). (2.10). The last equality follows from representation of the scale factor h in terms of a Fourierseries in σ.

2 Jul 2024: To remedy this, we modify our definition of ⟨ϕ|B for anti-linearoperators to. ... B†B = BB† = 1. – 17 –. Anti-Unitary Operators Conserve Probability.

8 Mar 2004: say. Then. Θ′′ = λΘ. = Θ =. A Bθ λ = 0A cos λθ B sin λθ λ 6= 0To obtain a sensible physical solution, replacing θ by θ 2π should give the same ... Θ =. A Bθ n = 0A cos nθ B sin nθ n 6= 0Returning to (2),.

23 Oct 2014: ui(0) = sin( πxi. b. ), u′i(0) =. π. b. gd(xi) cos. ( ... b. gd(xN ) cos. ( πxNb. ))T,. (39). where U (t) denotes the N -dimensional vector with entries ui(t) u(xi, t),.

10 Apr 2011: In a coordinate system moving at the speed of the colony, theradial and polar velocities are ur U[(c x3)cos A(x, )]and u U[(d x3)sin B(x, )], ... B 254, 107–113.12. Berg, H. C. & Purcell, E. M. (1977) Biophys.

22 Jul 2019: u =e cos(2py). I0() 1, v = 0. (3.2a,b). Given this fixed, known velocity the concentration simply evolves via the advection–diffusion equation (2.7). ... u(x, y, t) =e cos(2py)Rc0(x,y,t). Z 1. 0e cos(2ps)Rc. 0(x,s,t) ds 1.

22 Nov 2017: rA A O ‘( cos u1, sin u1) ð4:1Þ. and. rB B O ‘( cos u1 þ cos u2, sin u1 þ sin u2), ð4:2Þ. ... 0 its magnitude andt̂ ( cos u2, sin u2) the unit tangent vector joining A and B.Under the assumption of creeping flow, the drag forces areFA 2 zvA

13 Dec 2018: Example: if a and b are vectors then a. b is a scalar. ... b/|a| |b| = cos θ is a scalar, so that the angle between vectors is unaffected by a changeof frame.

12 Oct 2020: Thus, the linear system Au = b can be written explicitlyin the block form. ... By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,.

1 Nov 2018: CA =. 0. B@cos sin 0sin cos 0. 0 0 1. ... X cos Y sin. Again, U () must act on X this way if it is indeed to correspond to arotation.

4 Aug 2015: where. Vn = (An Bny)eny sin nξ (Cn Dny)eny cos nξ.(11). ... 3), is given by. rs = (δ ρ cos θ )x ρ sin θ y z z, δ = b sin k(z ct ),(18).

3 Oct 2008: X′′ k2X = 0. T = A sin(ckt) B cos(ckt). X = C sin(kx), k =nπ. ... L, n = 1, 2, 3,. 46. Elementary solution:. u =. [. A sin. (. nπct. L. ). B cos. (. nπct.

7 Jan 2004: a aT b bT = (cos θ a sin θ b) (cos θ a sin θ b)T. ... cos θ b sin θ a) (cos θ b sin θ a)T , θ R, a, bRn̂, (5.4).

8 Jul 2021: θ ),. ξx (t ) = ξ0 {[A(R) B(R) cos2 θ ]eiωt } and ξy (t ) = ξ0 {B(R) cos θ sin θ eiωt }. (17). B. Low-frequency limit of Lissajous figures. The ... 1 cos2θ )2 ξ2x. 2 tan θ cos φ. 1 cos2θ ξxξy 1.

3 Dec 2022: with. @B. @t=. @E. @x= kE0 cos(kx! t). We find. B =E0c. ... E = cos(k x! t) sin(k x! t). with Maxwell equations ensuring that k = k = 0.

9 Nov 2014: 2. V (t) =µπ2c2. 4L. n=1. n2[An cos. (nπct. L. ) Bn sin. ... m=1. i=1. [Cmi cos mθ Dmi sin mθ] cos(kmict) Jm(kmir). (5.32).

7 Oct 2020: Thus, the linear system Au = b can be written explicitlyin the block form. ... By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,.

16 Sep 2022: 00 = 30. r) = A. B. r2. The first term is just a constant rotation, while. ... 21. R22 R21. and B = (1 2)R21R. 22. R22 R21.

27 May 2021: θ ),. ξx (t ) = ξ0 {[A(R) B(R) cos2 θ ]eiωt } and ξy (t ) = ξ0 {B(R) cos θ sin θ eiωt }. (17). B. Low-frequency limit of Lissajous figures. The ... 1 cos2θ )2 ξ2x. 2 tan θ cos φ. 1 cos2θ ξxξy 1.

19 Oct 2013: be similar to that of the spherical harmonicP4(cos ), and so we adopt the estimate L 4:5. ... Then equations(1) and (2) imply T=T r2c 2i =gL and juj 2g T=2N2T , where we have takensin cos 21=2.

17 Jun 2004: 2, for f (x) = cos 10x, g(x) = x,[a, b] = [0, 1] and 400 < ω < 500. ... Thus, consider Fig. 2.1, where we have plottedthe error for f (x) = cos 10x, g(x) = x, a = 0, b = 1.

2 Sep 2012: sider the circular trajectory [see Fig. 1(b)]. R ðÞ bðcos; sin; 0Þ: (10)For this trajectory, we have jR0ðÞj b and tðÞ ð sin; cos; 0Þ. ... Whenthey are not parallel, the phase dependence is proportionalto RðÞ. For example, for perpendicular

22 Feb 2016: n 1)(n (n 1) cos x). (2n 1)(2n 1). xπ. 1. ... H′3 H′′′3 = G1. 1. Pn1 (1 cos x)n1, H3 Sx.

5 Feb 2020: Sébastien Michelin, a Simon Game,b Eric Lauga, c Eric Keaveny b andDemetrios Papageorgiou b. ... usingsinh(iz) = i sin z and cosh(iz) = cos iz.

9 Oct 2020: S =. α β. β α. β. β α.  ,are given by λk = α 2 i β cos kx , and wk = (i. ... λk (A) =λk (C ). λk (B )=. 1 12 i µ cos kx.

26 Oct 2020: y ′′ a(x )y ′ b(x )y = f (x ), y (1) = y (1), (6). ... For instance: choosinga(x ) = f (x ) = cos πx , b(x ) = sin 2πx (which incidentally evenleads to a sparse matrix) we get.

26 Jul 2013: 3) leads to 2p = 0. A solutionof the harmonic equation for the pressure is, at the first order inkb, p = p1(r,θ) (A cos s B sin s) with s = ... xi = x0 b cos β sin (s φ) , (34)yi = y0 b sin β sin (s φ).

10 May 2022: e1 = cos χ [cos θ (cos φex sin φey) sin θez] sin χ [sin φex cos φey] , (1). ... e2 = sin χ [cos θ (cos φex sin φey) sin θez] cos χ [sin φex cos φey] , (2).

31 May 2024: Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2019. General RelativityUniversity of Cambridge Part III Mathematical Tripos. David Tong. Department of Applied Mathematics and Theoretical Physics,. Centre for Mathematical Sciences,.

26 Feb 2008: Then. |R r|2 = (R r). (R r) = R2 r2 2Rr cos θ. ... u =GM. h2+ A cos(θ θ0). where A and θ0 are constants.

10 Jun 2016: ϕ=0. z=a2ρ2. z=a2ρ2. ρ dρ dϕ dz = 2π. a. b. ... Either way, the formula yields b. af(x) dx =. β. αf(x(u))dx.

26 Jan 2017: ûc(λ,t) =. 0. cos(λx)u(x,t)dx, û0c(λ) =. 0. cos(λx)u0(x)dx. (c) Equation (3.14) immediately implies that u(x,t) satisfies the heatequation. ... Example Consider the heat equation on the half line with. u(x, 0) = ea2x, u(0, t) = cos(bt), a,b real

22 Dec 2014: eπ2. sinh(π) j = 0,. j cos( jt)sin( jt)π2. j21. j Z+. ... a = 0, b = π, (t [0,π],q(t) = 20 cos(2t)), α1 6= 0, β1 6= 0, α2 = β2 = 0,.

2 Sep 2012: B. Liverpool, Phys. Rev. E 74, 061913 (2006).[10] D. Saintillan and M. ... E. Paullet and G. B.Ermentrout, SIAM J. Appl. Math. 54, 1720 (1994).

3 Oct 2016: κ3(ω) = 2/. [1 exp. (log4(1 ω). )];. d. κ4(ω) = cos(. ... The function κ4 is a special case of. κa,b(ω) = cos(.

10 Apr 2011: as a Fourier series:. R(z) = R0(. 1. 0dq b(q) cos(qz φ(q)). ... Our aim is to expand up to second orderin the coefficients b(q) characterizing the deformation.

24 Jun 2013: FIG. 2 (color online). Steady-state circulation in highly concentrated B. subtilis droplets. ... 2(e)]. The averagevalue b4m is slightly smaller than the length ‘5mof B.

4 May 2010: eη cos ωt = I0(η) 2. m=1. Im(η) cos mωt, (4). ... Alternatively, if wedefine the potential. H(p,q) = 12p2. g. cos q σ.

5 Feb 2020: PnmdyðfC0. sin f. cos f. 0. 0BBB@. 1CCCAeimfdf:. (29). The components of the angular velocity are. ... 1). Fig. 4 Concentration distributions: quantitative comparison of theory(solid curves) and simulations (symbols) for the asymmetric (black circles,k0

1 Dec 2009: 01. ]. µ cos ωt, (5.1). thereby, using our notation,. h(x) =. [0 1k b. ]. x, f(x) =. [01. ]. In this case, since the system is already linear, the matrixU is ... Putting everything together, we have. ψ1(t) = p1,0 µf(p0,0) cos ωt.

21 Oct 2020: By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,. ... 25 / 28. Splitting of inhomogeneous systems. Note that our analysis should accommodate b = b(t ), sinceboundary conditions might vary in time!

28 Oct 2020: For instance: choosinga(x ) = f (x ) = cos πx , b(x ) = sin 2πx (which incidentally evenleads to a sparse matrix) we get. ... Tn(x ) := cos n arccos x, x [1, 1],. or, in a more instructive form,.

13 Jul 2009: d&e 2( 1 - a2)8 2ar ------= dr ylr 4 -QB & (J&B - 1). ... Substituting our solution for the streamfunction, we find a stress. fr,, = -p Kr-2’“-2)(3 - n>n cos%ze,.

28 Feb 2012: 3.4) wx =DW(z). exp(ζt) cos( x),. where D denotes a differentiation with respect to z. ... cos. ((m 1)κz. )]}= 0,(3.22). where A is given by (3.11) and B, a modified inertial parameter, is defined as.

18 May 2022: 2A0A(cos(2ϕ1) cos(2ϕ2)) 0. 0 0. ) O((πN)1), (S37). Ĉ2 = 1. ... 0)2 ) η cos(ϕ. (0)2 ), (S46). and is obtained by multiplying Eq.

23 Apr 2021: b = R sin. 2. 2. = R cos. 2. If b > R, clearly there is no scattering. ... incident(r) = eikz = eikr cos. is also a solution to the Schrödinger equation.

23 Aug 2023: c)f = t2 t sin t t2 , f′ = (2 cos t)t sin t 6 2t as t. ... cos(β. 2). β. I(x) =. (bc)λβ(ac)λβ. f(c. y. λβ. )exp. (iλψ.

9 Oct 2017: qτmcE). cosh(qτmcE). 0 00 0 cos. (qτmB). sin(qτmB). 0 0 sin(qτmB). ... ẏ(0) cos. (qτmB)ż(0). ]z(τ) =. m. qB. [cos(qτmB)ẏ(0) sin.

13 Oct 2014: b) Now consider an almost ideal FJC with N monomers of type a. ... Show that these can be written as. W(n)(x) = A cos(k(n)x) B sin(k(n)x) D cosh(k(n)x) E sinh(k(n)x) ,. and